Polynomials for Crystal Frameworks and the Rigid Unit Mode Spectrum

Details Polynomials for Crystal Frameworks and the Rigid Unit Mode Spectrum Polynomials for Crystal Frameworks and the Rigid Unit Mode Spectrum

2011-02-14

arxiv.org/abs/1102.2744v2

Mathematics

Combinatorics

Revised convention for the matrix function of a crystal framework

Scientific paper

Two derivations are given for a matrix-valued function $\Phi_\C(z)$, defined on the $d$-torus, that can be associated with a discrete, translationally periodic bar-joint framework $\C$ in $\bR^d$. The rigid unit mode (RUM) spectrum of $\C$ is defined in terms of the phases of phase-periodic infinitesimal flexes and is identified in terms of the singular points of the function $z \to \rank \Phi_\C(z)$ and also in terms of the wave vectors of excitations with vanishing energy in the long wavelength limit. To a crystal framework $\C$ in Maxwell counting equilibrium we associate a unique multi-variable monic polynomial $p_\C(z_1,...,z_d)$ and for ideal zeolites the algebraic variety of zeros of $p_\C(z)$ on the $d$-torus determines the RUM spectrum. The matrix function is related to periodic "floppy modes" and their asymptotic order and an explicit formula is obtained for the number of periodic floppy modes for a given supercell. The crystal polynomial, RUM spectrum and the mode multiplicity are computed for a number of fundamental examples. In the case of certain zeolite frameworks in dimensions 2 and 3 direct proofs are give to show the maximal floppy mode property (order $N$). In particular this is the case for the cubic symmetry sodalite framework and for some novel two-dimensional zeolite frameworks.

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